Affiliations: Image and Pattern Analysis Group, University of Heidelberg, Speyerer Str. 6, 69115 Heidelberg, Germany. petra@math.uni-heidelberg.de | Image and Pattern Analysis Group, University of Heidelberg, schnoerr@math.uni-heidelberg.de | Institute of Aerodynamics and Flow Technology, German Aerospace Center, Bunsenstr. 10, 37073 Göttingen, Germany. andreas.schroeder@dlr.de
Note: [] Support by the German Research Foundation (DFG) is gratefully acknowledged, grant SCHN457/11. Address for correspondence: Image and Pattern Anal. Gr., Univ. of Heidelberg, Speyerer Str. 6, 69115 Heidelberg, Germany
Abstract: We analyze representative ill-posed scenarios of tomographic PIV (particle image velocimetry) with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine: (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.
Keywords: compressed sensing, underdetermined nonnegative linear systems, sparsity, large deviation, tail bound, limited angle tomography, TomoPIV
